(a) Field of the Invention
Broadly speaking, this invention relates to the selective deflection of ultrasonic acoustic beams. More particularly, in a preferred embodiment, this invention relates to the selective deflection of ultrasonic acoustic beams in piezoelectric solids by the use of externally addressable electronic signals.
(B) Discussion of the Prior Art
The steering of ultrasonic acoustic beams has application to a wide variety of devices such as acousto-optic deflectors, modulators and Q-switches; fiber optic and integrated circuit couplers; piezoelectric crystal filters, resonators, and delay lines, as well as to acoustic imaging, switching and multiplexing devices. Unfortunately, heretofore, existing devices for deflecting an acoustic beam have not been entirely satisfactory due to the size and cost of the equipment which is necessary to effect such changes. What is clearly needed is a device which is inexpensive and efficient, compatible with microcircuitry, all solid-state, rugged, and radiation resistant. Such a device has fortunately been achieved in the invention to be disclosed and claimed below,
The distinction between elastic and electroelastic waves in quartz has been experimentally demonstrated. The apparent inequality of the measurement of the two elastic constants, c.sub.14 and c.sub.56, is explained by taking into account piezoelectric effects. The stiffened elastic coefficients, c.sub..alpha..beta..sup.kD, have been shown to determine the propagation velocity of elastic-type waves in piezoelectric materials. (In the following discussion, Greek letter subscripts take on the values 1 to 6; Latin subscripts, 1 to 3). Subsequently, the c.sub..alpha..beta..sup.kD were shown to determine other elastic properties of unbounded piezoelectrics.
The relation between stiffened and unstiffened propagation velocities along a common wave normal in an unbounded piezoelectric has been considered analytically. For propagation directions, modes, and materials wherein the piezoelectric coupling is large and the stiffening contribution to the unstiffened velocity, v, is also large, the three stiffened characteristic values, .lambda..sub.i .ident. .rho.v.sub.i.sup.2 (.rho. is the material density) are in general not closely and simply related to the three unstiffened ones, .lambda..sub.j .ident. .rho.v.sub.j.sup.2 ; however, when the piezoelectric contribution is .lambda..sub.j is small, then each stiffened mode corresponds to a specific unstiffened one, a correspondence which is usually taken for granted. Clearly, when such a correspondence exists, a direct comparison of any particular stiffened and unstiffened quantity is meaningful. In the following discussion, the orientation of the normal-mode or characteristic vectors U.sub.i for stiffened or electroelastic waves relative to those S.sub.j for unstiffened or elastic waves with parallel wave fronts will be considered. (The vectors and the sets of axes they define are hereinafter referred to as "stiffened" or "unstiffened".)
Inherent in this analysis is the mutual orthogonality of the three vectors of each set. On the assumption that the U.sub.i and S.sub.j have a common origin, the two sets are rotated with respect to each other. This is shown after obtaining expressions for the U.sub.j in the coordinate frame of the S.sub.j. The form of these expressions shows that the matrix representing the rotation is compatible with an infinitesimal rotation about a unique direction. The magnitude and direction of rotation, which are functions of the direction of propagation, depend in detail upon the differences between pairs of unstiffened characteristic values and upon the components of the piezoelectric stiffening vector in the unstiffened coordinate frame, the latter ultimately depending upon the piezodielectric coefficients. When one of the modes is unstiffened, the rotation is about the unstiffened axis. The notation used is explained as introduced.
The propagation of a plane elastic wave in a semi-infinite piezodielectric is described by EQU (.GAMMA..sub.ij.sup.kD - .lambda. .delta..sub.ij)V.sub.j.sup.kD. (1)
(summation over repeated subscripts of factors is understood throughout except as indicated.) .GAMMA..sub.ij.sup.kD .ident. .GAMMA..sub.ij + .XI..sub.i .XI..sub.j /.xi..sup.s where .GAMMA..sub.ij .ident. c.sub.irjs.sup.E l.sub.r l.sub.s is the sum of products of the unstiffened elastic constants with direction cosines of the wave normal, .XI..sub.i .ident. e.sub.ist l.sub.s l.sub.t is the sum of products of the piezoelectric constants and direction cosines, and .epsilon..sup.s .ident. .epsilon..sup.vu l.sub.v l.sub.u (.epsilon..sub.vu are the dielectric permittivity tensor components). The .GAMMA..sub.ij are the elastic stiffnesses and the .XI..sub.i .XI..sub.j /.epsilon..sup.s are the piezoelectric stiffening contribution to .GAMMA..sub.ij. Equation (1) is written in the coordinate system defined by the crystallographic axes.
The analysis is simplified in a coordinate system in which .GAMMA..sub.ij is diagonal. The matrix S defining this frame obtains more (.GAMMA..sub.ij - .lambda..delta..sub.ij)S.sub.j =0. .GAMMA..sub.ij is transformed by the well-known rule S.GAMMA.S.sup.-1 to .lambda.I, the diagonal matrix .lambda..sub.i .delta..sub.ij ;.XI..sub.i .XI..sub.j /.epsilon..sup.s by S.XI..XI..sup.-1 S.sup.-1 /.epsilon..sup.s EQU ((.lambda..sub.i -.lambda.) .delta..sub.ij + Y.sub.i Y.sub.j) U.sub.j = 0. (2)
(No summation over i in .lambda..sub.i .delta..sub.ij.) When the stiffening contribution is small, perturbation type solutions for .lambda. result: EQU .lambda..sub.i = .lambda..sub.i + .DELTA..sub.i where (3) EQU .DELTA..sub.i = Y.sub.i.sup.2. (4)
With this brief introduction, the U.sub.j may now be evaluated. Before doing so, note that the real symmetric nature of the coefficients of U.sub.j in Eq. (2) requires that the U.sub.i be mutually orthogonal. consequently, the U.sub.i, which are taken as being normalized, may either be rigidly rotated or remain unchanged with respect to the (unstiffened) basis. Also, the correspondence of the i.sup.th stiffened axis with the i.sup.th unstiffened axis is insured by Eq. (3), which expresses the correspondence between .lambda..sub.i and .lambda..sub.i, and the validity of Eqs. (3) and (4) is based upon Y.sub.i.sup.2 /.lambda..sub.i &lt;&lt;1. These remarks suggest that the rotation be an infinitesimal one. Accordingly, the U.sub.i are expected to be of the form (.about.1, .about.0,.about.0) and expressions of this type are sought. (The symbols .about.1 and .about.0 signify quantities nearly equal to 1 and 0).
The U follow from Eq. (2), which is written in expanded form below: EQU (.lambda..sub.1 -.lambda.)U.sub.1 +Y.sub.1 Y.sub.2 U.sub.2 +Y.sub.1 Y.sub.3 U.sub.3 =0, (5a) EQU Y.sub.2 Y.sub.1 U.sub.1 +(.lambda..sub.2 -.lambda.)U.sub.2 +Y.sub.2 Y.sub.3 U.sub.3 =0, (5b) EQU Y.sub.3 Y.sub.1 U.sub.1 +Y.sub.3 Y.sub.2 U.sub.2 +(.lambda..sub.3 -.lambda.)U.sub.3 =0. (5c)
After eliminating, say, U.sub.3 from Eqs. (5a) and (5b), and U.sub.2 from Eqs. (5a) and (5c), one obtains EQU U.sub.1 =AY.sub.1 /(.lambda.-.lambda..sub.1), U.sub.2 =AY.sub.2 /(.lambda.-.lambda..sub.2), U.sub.3 =AY.sub.3 /(.lambda.-.lambda..sub.3). (6)
normalizing U gives A as:
1/A.sup.2 =Y.sub.1.sup.2 /(.lambda.-.lambda..sub.i).sup.2 +Y.sub.2.sup.2 2/(.lambda.-.lambda..sub.2).sup.2 +Y.sub.3.sup.2 /(.lambda.-.lambda..sub.3).sup.2. (7)
particularizing .lambda.=.lambda..sub.1 =.lambda..sub.1 +Y.sub.2.sup.2, EQU a.sub.1 =y.sub.1 /{1+y.sub.1.sup.2 y.sub.2.sup.2 /(.lambda..sub.1 -.lambda..sub.2).sup.2 +y.sub.1.sup.2 y.sub.3.sup.2 /(.lambda..sub.1 -.lambda..sub.3).sup.2 }.spsp.1/2 (8)
within this approximation, differences such as (.lambda..sub.1 -.lambda..sub.2) and (.lambda..sub.1 .lambda..sub.3) become (.lambda..sub.1 -.lambda..sub.2) and (.lambda..sub.1 -.lambda..sub.3) when .lambda..sub.1, .lambda..sub.2, and .lambda..sub.3 are not nearly equal to each other. Accordingly, Eq. (8) yields A.sub.1 =Y.sub.1 approximately and symmetry in the indices gives EQU A.sub.i =Y.sub.i ( 9)
Substituting .lambda..sub.1 for .lambda. in Eq. (6) determines U.sub.1. Terms such as (.lambda..sub.1 -.lambda..sub.2) and (.lambda..sub.1 -.lambda..sub.3) occur and are again replaced by (.lambda..sub.1 -.lambda..sub.2) and (.lambda..sub.1 -.lambda..sub.3). Finally, U.sub.1 =(.about.1, Y.sub.1 Y.sub.2 /(.lambda..sub.1 -.lambda..sub.2), Y.sub.1 Y.sub.3 (.lambda..sub.1 -.lambda..sub.3)). (10)
u.sub.2 and U.sub.3 are similarly found and one obtains ##EQU1## The array is in the form of an infinitesimal rotation matrix; the rotation "vector" is EQU d.OMEGA.=(Y.sub.2 Y.sub.3 /(.lambda..sub.2 -.lambda..sub.3), Y.sub.3 Y.sub.1 /(.lambda..sub.3 -.lambda..sub.1), Y.sub.1 Y.sub.2 /(.lambda..sub.1 -.lambda..sub.2)) (12)
any simple relation of the rotation axis to some vector or other property of a crystal is not obvious. Its scalar product with the stiffening vector Y is EQU d.OMEGA..multidot.Y=Y.sub.1 Y.sub.2 Y.sub.3 (1/(.lambda..sub.1 -.lambda..sub.2)+1/(.lambda..sub.2 -.lambda..sub.3)+1/(.lambda..sub.3 -.lambda..sub.1)), (13)
which in general is neither 0.degree. nor 90.degree.. When one of the Y.sub.i =0, the derivation breaks down because multiplication by 0 would then have been performed in arriving at Eq. (6). In this case, for example, Y.sub.3 should be set equal to zero in Eqs. (5a), (5b), and (5c) before beginning the elimination process. The result, however, is compatible with Eqs. (12) and (13). The rotation is about the (0,0,1) axis, leaving U.sub.3 =(0,0,1); .lambda..sub.3 =.lambda..sub.3 and to higher order approximation .lambda..sub.+ =.lambda..sub.1 -Y.sub.1.sup.2 Y.sub.2.sup.2 /(.lambda..sub.1 -.lambda..sub.2) and .lambda..sub.- =.lambda..sub.2 +Y.sub.1.sup.2 Y.sub.2.sup.2 /(.lambda..sub.1 -.lambda..sub.2).
So far the information obtained pertains to an initial state and a final state, states which are close to each other and connected by Y. We consider next Y to be a continuously variable parameter, choose a particular way in which to vary it, and explore what happens in between the initial and final states. Y is to be varied in such a way that it remains parallel to itself while its magnitude increases or scales from 0 (the initial state to .vertline.Y.vertline. (the final state); Y=fY, where 0.ltoreq.f.ltoreq.1.
Within the small coupling approximation, Eq. (12) applies, d.OMEGA. is homogeneous in products of Y.sub.i to second degree. Its magnitude varies as f.sup.2 and its direction is unchanged, provided .lambda..sub.1, .lambda..sub.2, and .lambda..sub.3 are assumed to remain constant as f varies. This assumption is valid as this is precisely the meaning of the small coupling approximation: Eqs. (3) and (4) apply and Eq. (4) becomes .DELTA..sub.i =f.sup.2 Y.sub.i.sup.2. Consequently, as f is increased continuously to 1, the unstiffened vectors may be considered to be continuously and rigidly rotated about a unique axis to their final position. Each vector prescribes a cone about d.OMEGA..